Many Japanese are viewing the economic summit just concluded here as a failure for their country, posing some serious questions for the political future of the man who spoke for them, Prime Minister Yasuhiro Nakasone.

Nakasone is widely believed to nurse long-shot hopes for an unprecedented ex-tension in office after his current term expires in the fall. Public perceptions that he botched the summit could present serious obstacles to any such plan.

The summit dealt with many issues. But Japanese have tended to judge it by only one, whether it would stop a six-month, 45 percent rise of the value of the yen, which is beginning to cut painfully into Japanese industry by making its products more expensive to foreign buyers.

During summit sessions, Nakasone argued that the yen has gone intolerably high and something must be done. Only the West Germans gave his concerns much note. Others said the strong yen was long overdue and crucial to correcting trade imbalances.

Word of this raced around the world and the yen soared again, reaching on the summit's final day a postwar record strength of 165.20 to the dollar.

Many Japanese do see positive steps coming from the summit, particularly a commitment to closer cooperation and consultation in politics and economics. But the pluses tend to be abstract and the minuses specific -- a matter of money.

By some accounts, no one could have done much better than Nakasone did. "People expected that we could stop the appreciation of the yen or at least reverse the trend," said Bunroku Yoshino, chairman of the Institute for International Economic Studies, a major Japanese think tank. "That was of course a futile effort from the beginning. But that's what was expected of him."

Opposition parties have issued predictable condemnations of the prime minister's performance. More significantly, some senior figures in the ruling Liberal Democratic Party and leaders of big business have complained over the outcome, by inference criticizing Nakasone.

"I am unsatisfied," Takashi Ishihara, chairman of the Japan Committee for Economic Development, an influential association of business leaders, told a newspaper. "I wish that they had stressed more effective joint intervention in the currency markets."

Those fearing that a strong yen means unemployment could turn to Nakasone's rivals for the prime ministership, say analysts. Nakasone is in his fourth year in office. The summit was important to sustain his reputation for facility in dealing with foreigners.

Nakasone can make jokes in English and has established a first-name aquaintance with President Reagan that is known here as "the Ron-Yasu relationship."

Political commentators say Nakasone felt that scenes of him chatting amicably with Reagan and the five other world leaders, followed by communiques that would clearly be labeled a "success" for Japan, would boost his political stock considerably.

That would strengthen his hand to call elections for the lower house of the Diet, or legislature, this summer -- at the same time as an already scheduled upper house vote. A strong showing therefore could set the stage for wangling an unprecedented extension in office this fall on expiration of his second, and by party rules, last term.

But many Japanese may think Nakasone played the role of toady, not confidant of the foreign leaders. His vaunted friendship with Reagan seemed to get him nowhere.

Nakasone, calling the summit "full of substance" in a press conference yesterday, said, "Japan's weight and status has increased but at the same time so has its responsibilities."

But his government already has gone to work on damage control. New aid programs for small and medium industries, which have been especially hard hit by the yen's rise, are to be announced later this week, and there is talk of new public works spending to prop up the economy.

"The key thing for the government and Nakasone will be what they can do within the next two weeks concerning countermeasures for the domestic economy," said Jushiro Komiyama, a ruling party legislator.

Frightened by shrill condemnation of its trading practices and burgeoning trade surpluses ($46 billion in 1985 by Japanese statistics), Japan agreed last September to join the United States, Britain, France and West Germany to intervene in markets to bring down the the dollar and push up the yen.

This made Japanese goods more costly to foreign buyers and reduced exports. But it was viewed here as better than risking protectionism abroad. The dollar was worth about 240 yen when the program began; it has slid steadily to the 165 level this week.

One of the summit's major initiatives was to move toward a more formal management of exchange rates to prevent disruptive ups and downs. Meeting the press yesterday, Nakasone depicted this as a tool by which the "yen problem" might be solved. But public statements by the United States and other countries made clear they did not intend to apply it that way.

Nakasone also has taken fire for bending too readily to foreign pressure on Libya. Japan pursues a generally timid foreign policy, especially when its economic interests are at stake. After the summit closed, Nakasone admitted publicly that he was forced to yield to the other six summit leaders' insistence that Libya be singled out. "Except for Japan, all of the countries favored naming Libya because evidence of its involvement, as a government, was clear," he said. "As chairman, I was in a position in which I felt I must accept the opinions of the others."